Moving in a straight line
I don't know if there is a way to achieve this with the green arrows, but it is done easily using the movement hotkeys. If you press G, the vertices will move freely. If you press x, y, or z after that, they will only move in that direction. But if you press G + Z + Z, it will not move in the global z direction, but in the local z direction, which is what you want here. For example, in the image below, I have a rotated cube, and I move 1 face not upwards, but in the direction that is it facing, by pressing z twice after pressing g.



I hope this helps!